It has been demonstrated that differences in shear moduli are related to differences in principal stresses in a homogeneously stressed rock. There are two independent difference equations relating the three shear moduli C44, C55, and C66, and three unknowns—the maximum and minimum horizontal stresses, and an acoustoelastic coefficient defined in terms of two rock nonlinear constants (C144 and C155). Consequently, two independent equations relating four unknowns have to be solved.
To overcome these limitations, the applicability of shear moduli difference equations is generalized in the presence of known stress distributions caused by the presence of a borehole. Near-wellbore stress distributions are known from the theory of elasticity that is valid for rock stresses less than the rock yield stress. Dipole shear moduli C44 and C55 change as we approach near-wellbore region where the far-field stresses change to borehole cylindrical stresses. Radial and azimuthal variations of these stresses are known from the linear elasticity. Since these difference equations contain two unknown nonlinear constants C144 and C155, four unknowns need to be solved. However, two more difference equations can be formed that relate changes in the dipole shear moduli C44 and C55 at two radial positions to the corresponding changes in borehole cylindrical stresses. These borehole stresses can be expressed in terms of the three formation principal stresses. One of the equations relates the difference between [C55(r/a=far)−C55(r/a=near)] to corresponding stresses at these two radial positions normalized by the borehole radius a. The second equation relates the difference between [C44(r/a=far)−C55(r/a=near)] to the stresses at these two radial positions. Radial variations of shear moduli C55 and C44 are obtained from the dipole Shear Radial Velocity Profiling (SRVP) algorithm using the fast- and slow-dipole dispersions.
Therefore, these four equations can be solved to obtain the maximum and minimum horizontal stresses and the nonlinear constants C155 and C144 referred to a local reference state. Higher-order coefficients of nonlinear elasticity C144, C155, and C456 are also used to calculate stress coefficients of shear velocities from an acoustoelastic model of wave propagation in prestressed materials (“Third-order constants and the velocity of small amplitude elastic waves in homogeneously stressed materials”, by R. N. Thurston and K. Brugger, Physical Review, vol. A 133, pp. 1604-1610, 1964; “Elastic waves in crystals under a bias”, by B. K. Sinha, Ferroelectrics, vol. 41, pp. 61-73. 1982).
Various devices are known for measuring formation characteristics based on sonic data. Mechanical disturbances are used to establish elastic waves in earth formations surrounding a borehole, and properties of the waves are measured to obtain information about the formations through which the waves have propagated. For example, compressional, shear and Stoneley wave information, such as velocity (or its reciprocal, slowness) in the formation and in the borehole can help in evaluation and production of hydrocarbon resources. One example of a sonic logging device is the Sonic Scanner® from Schlumberger. Another example is described in Pistre et al., “A modular wireline sonic tool for measurements of 3D (azimuthal, radial, and axial) formation acoustic properties, by Pistre, V., Kinoshita, T., Endo, T., Schilling, K., Pabon, J., Sinha, B., Plona, T., Ikegami, T., and Johnson, D.”, Proceedings of the 46th Annual Logging Symposium, Society of Professional Well Log Analysts, Paper P, 2005. Other tools are also known. These tools may provide compressional slowness, Δtc, shear slowness, Δts, and Stoneley slowness, Δtst, each as a function of depth, z, where slowness is the reciprocal of velocity and corresponds to the interval transit time typically measured by sonic logging tools. An acoustic source in a fluid-filled borehole generates headwaves as well as relatively stronger borehole-guided modes. A standard sonic measurement system uses a piezoelectric source and hydrophone receivers situated inside the fluid-filled borehole. The piezoelectric source is configured as either a monopole or a dipole source. The source bandwidth typically ranges from a 0.5 to 20 kHz. A monopole source primarily generates the lowest-order axisymmetric mode, also referred to as the Stoneley mode, together with compressional and shear headwaves. In contrast, a dipole source primarily excites the lowest-order flexural borehole mode together with compressional and shear headwaves. The headwaves are caused by the coupling of the transmitted acoustic energy to plane waves in the formation that propagate along the borehole axis. An incident compressional wave in the borehole fluid produces critically refracted compressional waves in the formation. Those refracted along the borehole surface are known as compressional headwaves. The critical incidence angle θi=sin−1 (Vf/Vc), where Vf is the compressional wave speed in the borehole fluid; and Vc is the compressional wave speed in the formation. As the compressional headwave travels along the interface, it radiates energy back into the fluid that can be detected by hydrophone receivers placed in the fluid-filled borehole. In fast formations, the shear headwave can be similarly excited by a compressional wave at the critical incidence angle θi=sin−1 (Vf/Vs), where Vs is the shear wave speed in the formation. It is also worth noting that headwaves are excited only when the wavelength of the incident wave is smaller than the borehole diameter so that the boundary can be effectively treated as a planar interface. In a homogeneous and isotropic model of fast formations, as above noted, compressional and shear headwaves can be generated by a monopole source placed in a fluid-filled borehole for determining the formation compressional and shear wave speeds. It is known that refracted shear headwaves cannot be detected in slow formations (where the shear wave velocity is less than the borehole-fluid compressional velocity) with receivers placed in the borehole fluid. In slow formations, formation shear velocities are obtained from the low-frequency asymptote of flexural dispersion. There are standard processing techniques for the estimation of formation shear velocities in either fast or slow formations from an array of recorded dipole waveforms.
It is known that sonic velocities in rocks change as a function of porosity, clay volume, saturation, stresses and temperature. It is, therefore, necessary to invert only those velocity differences between two depths or radial positions that are largely due to stress changes and effects of any other contributing factors are eliminated. The underlying theory behind the estimation of formation stresses using borehole sonic data is based on acoustoelastic effects in rocks. Acoustoelasticity in rocks refers to changes in elastic wave velocities caused by changes in pre-stress in the propagating medium. Elastic wave propagation in a pre-stressed material is described by equations of motion for small dynamic fields superposed on a statically deformed state of the material. These equations are derived from the rotationally invariant equations of nonlinear elasticity (“Elastic waves in crystals under a bias”, by B. K. Sinha, Ferroelectrics, vol. 41, pp. 61-73. 1982; “Acoustoelasticity of solid/fluid composite systems”, by A. N. Norris, B. K. Sinha, and S. Kostek, Geophysical Journal International, vol. 118, pp. 439-446, August 1994). Equations of motion for pre-stressed isotropic materials contain two linear (λ and μ) and three nonlinear elastic stiffness constants (C111, C144, C155) in a chosen reference state together with the biasing stresses. A forward solution of equations of motion in pre-stressed materials yields plane wave velocities as a function of principal stresses in the propagating medium. An inversion algorithm estimates stresses in the propagating medium in terms of measured velocities.
All these techniques account for estimation of formation stresses using sonic data acquired in vertical wells, referred to an orthogonal trihedron including the vertical borehole axis. However, onshore, it is necessary to drill a deviated well to enter formations at selected locations and angles. This may occur because of the faulting in the region. It is also necessary to do this around certain types of salt structures. As a further example of onshore deviated drilling, there has been growing interest in providing surveys of wells that have been deviated from a vertical portion toward the horizontal.
In offshore production, once a producing formation has been located, it is typically produced from a centrally positioned platform. A single production platform is typically installed at a central location above the formation and supported on the ocean bottom. A production platform supports a drilling rig which is moved from place to place on the platform so that a number of wells are drilled. From the inception, most of the wells are parallel and extend downwardly with parallel portions, at least to a certain depth. Then, they are deviated at some angle. At the outer end of the deviated portion, vertical drilling may again be resumed. While a few of the wells will be more or less vertically drilled, many of the wells will be drilled with three portions, a shallow vertical portion, an angled portion, and a termination portion in the formation which is more or less vertically positioned. Therefore the need to provide formation stresses data in deviated wells is continuously increasing.